Random walk

Introduction

A random walk is a stochastic process which describes a path made of consecutive random steps.

Gaussian

In Gaussian random walk the steps follow a continous Gaussian distribution. We will look at two different types, the univariate and multivariate kind.

Univariate

A univariate Gaussian Random Walk, is a series of i.i.d. \(\mathcal{N}(0,1)\) random variables such that

\begin{align*} X_0&=0 \\ X_t&=X_{t−1}+\epsilon_t \end{align*}

Where \(t=1,2,\dots\) and \(\epsilon_t\) is a series of i.i.d. \(\mathcal{N}(0,1)\) random variables.

Let’s illustrate a simple univariate Gaussian random walk in Python, by plotting 1000 realisations.

import numpy as np

N = 1000
np.random.seed(23)
realisations = []
for i in range(N):
    realisations.append(np.cumsum(np.random.normal(size=100)))